Associative algebra

This article is about a standard (though not very rudimentary) definition in an area related to, but not strictly part of, group theory

Definition

An associative algebra over a base ring $R$ is defined as a ring $A$, along with the structure of a $R$-module to $A$.

In the particular case when $R$ and $A$ are both unital rings, this is equivalent to saying that we require an embedding of $R$ as a sub (unital ring) of $A$.

We typically studiy algebras over a field, which are just vector spaces over the field equipped with a suitable compatible multiplication.

Sometimes, we also look at the non-associative notion of algebra, where we do not assume associativity of the multiplication for $A$.

Related notions

• Coalgebra
• Bialgebra
• Hopf algebra
• Group algebra